An Hardy Estimate for Commutators of Pseudo-Differential Operators

Let T be a pseudo-differential operator whose symbol belongs to the Hörmander class Sᵐρ,δ with 0≤ δ< 1, 0< ρ≤ 1, δ ≤ ρ and -(n+1)< m ≤ - (n+1)(1-ρ). In present paper, we prove that if b is a locally integrable function satisfying _ balls\; B⊂ Rⁿ log(e+ 1/|B|)/(1+ |B|)^θ 1/|B|∫B |f(x)- 1/|B|∫B f(y) dy|dx < ∞ for some θ∈ [0,∞), then the commutator $[b,T]$ is bounded on the local Hardy space h¹( Rⁿ) introduced by Goldberg [9]. As a consequence, when ρ=1 and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen [21].